Synchronization in weighted complex networks: Heterogeneity and synchronizability

Synchronization in different types of weighted networks based on a scale-free weighted network model is investigated. It has been argued that heterogeneity suppresses synchronization in unweighted networks [T. Nishikawa, A.E. Motter, Y.C. Lai, F.C. Hoppensteadt, Phys. Rev. Lett. 91 (2003) 014101]. However, it is shown in this work that as the network becomes more heterogeneous, the synchronizability of Type I symmetrically weighted networks, and Type I and Type II asymmetrically weighted networks is enhanced, while the synchronizability of Type II symmetrically weighted networks is weakened.     

Modelling of weighted evolving networks with community structures

Many social and biological networks consist of communities–groups of nodes within which links are dense but among which links are sparse. It turns out that most of these networks are best described by weighted networks, whose properties and dynamics depend not only on their structures but also on the link weights among their nodes. Recently, there are considerable interests in the study of properties as well as modelling of such networks with community structures. To our knowledge, however, no study of any weighted network model with such a community structure has been presented in the literature to date. In this paper, we propose a weighted evolving network model with a community structure. The new network model is based on the inner-community and inter-community preferential attachments and preferential strengthening mechanism. Simulation results indicate that this network model indeed reflect the intrinsic community structure, with various power-law distributions of the node degrees, link weights, and node strengths.     

Weighted Evolving Networks with Self-organized Communities

In order to describe the self-organization of communities in the evolution of weighted networks, we propose a new evolving model for weighted community-structured networks with the preferential mechanisms functioned in different levels according to community sizes and node strengths, respectively. Theoretical analyses and numerical simulations show that our model captures power-law distributions of community sizes, node strengths, and link weights, with tunable exponents of ν≥1, γ>2, and α>2, respectively, sharing large clustering coefficients and scaling clustering spectra, and covering the range from disassortative networks to assortative networks. Finally, we apply our new model to the scientific co-authorship networks with both their weighted and unweighted datasets to verify its effectiveness.     

Weighted coupling for geographical networks: Application to reducing consensus time in sensor networks

Although many complex real-world networks are weighted, unweighted networks are used in many applications such as sensor networks. In this Letter it is shown using properly weighted networks the performance can be greatly enhanced by reducing the time necessary for the average consensus. Random geographical models are adapted as network models and a method based on mutually coupled phase oscillators is used for providing average consensus over the network. The consensus time is calculated by numerically solving the network's differential equations and monitoring the average error. The simulation results on some sample networks show that the consensus time is dramatically reduced when the proposed weights are used for the links of the underlying network.     

Scaling of weighted spectral distribution in weighted small-world networks

Many real-world systems can be modeled by weighted small-world networks with high clustering coefficients. Recent studies for rigorously analyzing the weighted spectral distribution(W SD) have focused on unweighted networks with low clustering coefficients. In this paper, we rigorously analyze the W SD in a deterministic weighted scale-free small-world network model and find that the W SD grows sublinearly with increasing network order(i.e., the number of nodes) and provides a sensitive discrimination for each input of this model. This study demonstrates that the scaling feature of the W SD exists in the weighted network model which has high and order-independent clustering coefficients and reasonable power-law exponents.     

Finding communities in weighted networks through synchronization

Community detection in weighted networks is an important challenge. In this paper, we introduce a local weight ratio scheme for identifying the community structures of weighted networks within the context of the Kuramoto model by taking into account weights of links. The proposed scheme takes full advantage of the information of the link density among vertices and the closeness of relations between each vertex and its neighbors. By means of this scheme, we explore the connection between community structures and dynamic time scales of synchronization. Moreover, we can also unravel the hierarchical structures of weighted networks with a well-defined connectivity pattern by the synchronization process. The performance of the proposed method is evaluated on both computer-generated benchmark graphs and real-world networks.     

Weighted and unweighted network of amino acids within protein

The information regarding the structure of a single protein is encoded in the network of interacting amino acids considered as nodes. If any two atoms from two different amino acids (nodes) are within higher cut-off distance of London-van der Waals forces, the amino acids are considered to be linked or connected. Several atoms of any amino acids in a protein may be within the above prescribed distance of several atoms of another amino acid resulting in possible multiple links between them. These multiple links are the basis of the weight of the connectivity in a protein network. Each protein has been considered as a weighted and an unweighted network of amino acids. A total of forty nine protein structures that covers the three branches of life on earth has been analyzed and several network properties have been studied. The probability degree and strength distributions of network connectivity have been obtained. It has been observed that the average strength of amino acid node depends on its degree. The results show that the average clustering coefficient of weighted network is less than that of unweighted network. It implies that the topological clustering is generated by edges with low weights. The power-law behavior of clustering coefficients of weighted and unweighted networks as a function of degree indicates that they have signatures of hierarchy. It has also been observed that the network is of assortative type.     

A model for the spread of rumors in Barrat–Barthelemy–Vespignani (BBV) networks

Rumor propagation is a typical form of social communication and plays a significant role in social life. In this paper, we studied the process of rumor propagation by accounting for the mechanism of forgetting in Barrat–Barthelemy–Vespignani (BBV) networks. First, we derived mean-field equations for rumor propagation based on the strength of the nodes in the propagation network. We then analyzed the stability of the model to determine whether a propagation threshold existed in the BBV networks. We also conducted numerical simulations of the BBV networks and found that rumors propagate more slowly in BBV networks than in unweighted networks. The numerical simulation results also demonstrated that as the forgetting rate increases, the rumor’s influence decreases in both BBV networks and unweighted networks. Finally, the simulation results confirmed that a threshold exists for rumor propagation in BBV networks, but that it was independent of the value of the stifling rate.     

Weighted Exponential Random Graph Models: Scope and Large Network Limits

We study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. Analogous to fundamental results derived for standard (unweighted) exponential random graph models in the work of Chatterjee and Diaconis, we derive limiting results for the structure of these models as the number of nodes goes to infinity. Our results are applicable for a wide variety of base measures including measures with unbounded support. We also derive sufficient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally we include a number of open problems to spur further understanding of this model especially in the context of applications.     

Emergence of communities in weighted networks

Topology and weights are closely related in weighted complex networks and this is reflected in their modular structure. We present a simple network model where the weights are generated dynamically and they shape the developing topology. By tuning a model parameter governing the importance of weights, the resulting networks undergo a gradual structural transition from a module-free topology to one with communities. The model also reproduces many features of large social networks, including the "weak links" property.     

Self-contained algorithms to detect communities in networks

The investigation of community structures in networks is an important issue in many domains and disciplines. In this paper we present a new class of local and fast algorithms which incorporate a quantitative definition of community. In this way the algorithms for the identification of the community structure become fully self-contained and one does not need additional non-topological information in order to evaluate the accuracy of the results. The new algorithms are tested on artificial and real-world graphs. In particular we show how the new algorithms apply to a network of scientific collaborations both in the unweighted and in the weighted version. Moreover we discuss the applicability of these algorithms to other non-social networks and we present preliminary results about the detection of community structures in networks of interacting proteins.Received: 7 November 2003, Published online: 14 May 2004PACS: 89.75.Hc Networks and genealogical trees - 87.23.Ge Dynamics of social systems - 87.90. + y Other topics in biological and medical physics     

Accuracy and precision of methods for community identification in weighted networks

Different algorithms, which take both links and link weights into account for the community structure of weighted networks, have been reported recently. Based on the measure of similarity among community structures introduced in our previous work, in this paper, accuracy and precision of three algorithms are investigated. Results show that Potts model based algorithm and weighted extremal optimization (WEO) algorithm work well on both dense or sparse weighted networks, while weighted Girvan–Newman (WGN) algorithm works well only for relatively sparse networks.     

On the rich-club effect in dense and weighted networks

For many complex networks present in nature only a single instance, usually of large size, is available. Any measurement made on this single instance cannot be repeated on different realizations. In order to detect significant patterns in a real-world network it is therefore crucial to compare the measured results with a null model counterpart. Here we focus on dense and weighted networks, proposing a suitable null model and studying the behaviour of the degree correlations as measured by the rich-club coefficient. Our method solves an existing problem with the randomization of dense unweighted graphs, and at the same time represents a generalization of the rich-club coefficient to weighted networks which is complementary to other recently proposed ones.     

Importance of tie strengths in the prisoner?s dilemma game on social networks

Though numerous researches have shown that tie strengths play a key role in the formation of collective behavior in social networks, little work has been done to explore their impact on the outcome of evolutionary games. In this Letter, we studied the effect of tie strength in the dynamics of evolutionary prisoner?s dilemma games by using online social network datasets. The results show that the fraction of cooperators has a non-trivial dependence on tie strength. Weak ties, just like previous researches on epidemics and information diffusion have shown, play a key role by the maintenance of cooperators in evolutionary prisoner?s dilemma games.     

Finding communities in directed networks by PageRank random walk induced network embedding

Community structure has been found to exist ubiquitously in many different kinds of real world complex networks. Most of the previous literature ignores edge directions and applies methods designed for community finding in undirected networks to find communities. Here, we address the problem of finding communities in directed networks. Our proposed method uses PageRank random walk induced network embedding to transform a directed network into an undirected one, where the information on edge directions is effectively incorporated into the edge weights. Starting from this new undirected weighted network, previously developed methods for undirected network community finding can be used without any modification. Moreover, our method improves on recent work in terms of community definition and meaning. We provide two simulated examples, a real social network and different sets of power law benchmark networks, to illustrate how our method can correctly detect communities in directed networks.     

Effects of the planarity and heterogeneity of networks on evolutionary two-player games

We study the effects of the planarity and heterogeneity of networks on evolutionary two-player symmetric games by considering four different kinds of networks, including two types of heterogeneous networks: the weighted planar stochastic lattice(a planar scale-free network) and the random uncorrelated scale-free network with the same degree distribution as the weighted planar stochastic lattice; and two types of homogeneous networks: the hexagonal lattice and the random regular network with the same degree k_0= 6 as the hexagonal lattice. Using extensive computer simulations, we found that both the planarity and heterogeneity of the network have a significant influence on the evolution of cooperation, either promotion or inhibition, depending not only on the specific kind of game(the Harmony, Snowdrift, Stag Hunt or Prisoner's Dilemma games), but also on the update rule(the Fermi, replicator or unconditional imitation rules).     

Link prediction based on a semi-local similarity index

Missing link prediction provides significant instruction for both analysis of network structure and mining of unknown links in incomplete networks. Recently, many algorithms have been proposed based on various node-similarity measures. Among these measures, the common neighbour index, the resource allocation index, and the local path index, stemming from different source, have been proved to have relatively high accuracy and low computational effort. In this paper, we propose a similarity index by combining the resource allocation index and the local path index. Simulation results on six unweighted networks show that the accuracy of the proposed index is higher than that of the local path one. Based on the same idea of the present index, we develop its corresponding weighted version and test it on several weighted networks. It is found that, except for the USAir network, the weighted variant also performs better than both the weighted resource allocation index and the weighted local path index. Due to the improved accuracy and the still low computational complexity, the indices may be useful for link prediction.     

The adaptive coupling scheme and the heterogeneity in intrinsic frequency and degree distributions of the complex networks

In the paper, we applied an adaptive principle to three kinds of complex networks as well as a random network within the context of the Kuramoto model. We found that the adaptive scheme could suppress the negative effect of the heterogeneity in the networks and the phase synchronization is enhanced obviously. The paper mainly investigates the adaptive coupling scheme in the small-world network, the scale-free network, and the modular network. Comparing with other weighted or unweighted static coupling schemes, the adaptive coupling scheme has a better performance in synchronization and communication efficiency, and provides a more realistic picture of synchronization in complex networks.     

Boolean networks with variable number of inputs (K)

We studied a random Boolean network model with a variable number of inputs K per element. An interesting feature of this model, compared to the well-known fixed-K networks, is its higher orderliness. It seems that the distribution of connectivity alone contributes to a certain amount of order. In the present research, we tried to disentangle some of the reasons for this unexpected order. We also studied the influence of different numbers of source elements (elements with no inputs) on the network's dynamics. An analysis carried out on the networks with an average value of K=2 revealed a correlation between the number of source elements and the dynamic diversity of the network. As a diversity measure we used the number of attractors, their lengths and similarity. As a quantitative measure of the attractors' similarity, we developed two methods, one taking into account the size and the overlapping of the frozen areas, and the other in which active elements are also taken into account. As the number of source elements increases, the dynamic diversity of the networks does likewise: the number of attractors increases exponentially, while their similarity diminishes linearly. The length of attractors remains approximately the same, which indicates that the orderliness of the networks remains the same. We also determined the level of order that originates from the canalizing properties of Boolean functions and the propagation of this influence through the network. This source of order can account only for one-half of the frozen elements; the other half presumably freezes due to the complex dynamics of the network. Our work also demonstrates that different ways of assigning and redirecting connections between elements may influence the results significantly. Studying such systems can also help with modeling and understanding a complex organization and self-ordering in biological systems, especially the genetic ones.     

权重分布对加权局域世界网络动力学同步的影响

加权网络可以更细致地刻画复杂系统中两节点之间的相互作用, 所以加权网络也比无权网络更接近真实的复杂系统. 改变权重分布来调节和改善复杂网络的性质也成为一种新的研究方法. 基于现有无权网络同步的概念, 应用特征值比R来衡量加权局域世界网络的同步能力, 发现权重分布对加权局域世界网络的同步能力有很重要的影响, 权重分布越均匀, 网络的同步能力就越大. 关键词： 权重分布 局域世界 加权网络 同步